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P!=NP proof (revised)

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The following is a snipet of the revised proof
https://sourceforge.net/projects/cscall/files/MisFiles/PNP-proof-en.txt/download

I think the idea of the proof should be valid and easy to understand. The rest
technical apart should be straightforward (could take pages or dozens of pages,
so ignored). But, anyway, something like the C/C++ description is still needed.
Can you find any defects?

OTOH, C/C++ can be the language for for proving math. theorems, lots easier than
TM language to handle and understand. Opinions?

--------
ℕℙ::= {q| q is a decision problem that a computer solves in O(2^|q|) steps using
       the following fnp algorithm template. q contains a verification dataset
       C, card(C)∈O(2^|q|), and a Ptime verification function v:C->{true,false}.
       If ∃c,v(c)=true, then the answer to problem q is true; otherwise, it is
       false.}

    // begin_certificate is a Ptime function that retrieves the first
    // Certificate element from the problem statement q. If this element does
    // not exist, it returns a unique and virtual EndC element.
    Certificate begin_certificate(Problem q);

    // end_certificate is a Ptime function that retrieves the element EndC from
    // the problem statement q.
    Certificate end_certificate(Problem q);

    // next_certificate is a Ptime function that retrieves the next element of
    // c from the problem statement q. If this element does not exist, return
    // the EndC element.
    Certificate next_certificate(Problem q, Certificate c);

    // v is a Ptime function. v(c)==true if c is the element expected by the
    // problem.
    bool v(Certificate c);

    bool fnp(Problem q) {
      Certificate c, begin, end;   // Declare the verification data variable
      begin= begin_certificate(q); // begin is the first verification data
      end= end_certificate(q);     // end is the false data EndC used to
                                   // indicate the end.
      for(c = begin; c != end;
          c = next_certificate(q, c)) { // At most O(2^|q|) steps.
                                        // next_certificate(c) is the Ptime
                                        // function to get the next
                                        // verification data of c
          if(v(c) == true) return true; // v: C->{true, false} is the polynomial
                                        // time verification function.
      }
      return false;
    }

    Since a continuous O(P) number of Ptime functions (or instructions) can be
    combined into a single Ptime function, if the complexity of each function is
    Ptime, and the smallest unit of complexity is also Ptime, then it's roughly
    the same. Any Ptime function can be added, deleted, merged, or split in the
    algorithm without affecting the algorithm's complexity. Perhaps in the end,
    only the number of decision branches needs to be considered.

    [Note] This definition of ℕℙ is equivalent to the traditional Turing machine
           definition of ℕℙ. The proof of equivalence is plain and lengthy, and
           not very important to most people, so it is omitted.

    [Note] According to the Church-Turing conjecture, no formal language can
           surpass the expressive power of a Turing machine (or algorithm) (i.e.
           the decisive operational process from part to whole). C language can
           be regarded as a high-level language for Turing machines, and as a
           formal language for knowledge or proof.

Problem Q::= Given plaintext a, ciphertext b, decoder d, and key length klen.
    The key is a Ptime program. Problem Q determines whether there exists a
    key c such that d(b,c)=a.

    Problem Q can be computed using the following C/C++ pseudo code as fnp by
    definition; therefore, Q∈ℕℙ.
    Plaintext a, ciphertext b, decoder d, and the length of key klen are all
    obtained from q:

    bool f(Problem q) {
      int MaxKey= ...                  // klen=maximum value of key (O(2^klen))
      for(int c=1; c<=MaxKey; ++i) {
        if(equ(d(b,c),a)) return true; // Polynomial-time verification
                                       // (If c is not a valid program, then d
                                       // returns a value such that equ test
                                       // result is false)
      }
      return false;
    }

    Since the key is a freely written program, Each decription algorithm (key)
    is essentially independent, and the given problem q may contain no
    information about the algorithm's logic (knowledge of one key cannot be
    used to deduce information about another).
    Therefore, at least O(2^|klen|) possible key values must be tested one by
    one. Thus, the complexity of problem q is O(2^N).
---------------

(Thanks to olcott, this proof was origianlly inspired by POO Halt, where the
 P-NP proof based on Halting Problem was found invalid)

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P!=NP proof (revised) wij <wyniijj5@gmail.com> - 2025-11-19 15:25 +0800
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